Problem: If $27^a=9$, and $9^b=27$, what is the value of $ab$?
Explanation: The first equation gives us that $3^{3a}=3^2$, so $a=\frac{2}{3}$, and the second equation gives us that $3^{2b}=3^3$, so $b=\frac{3}{2}$. The value of $ab$ is thus $\frac{2}{3}\cdot\frac{3}{2}=\boxed{1}$. Alternatively, we can take logarithms of both sides to find that $a=\log_{27} 9=\frac{\log 9}{\log 27}$ and $b=\log_9 27=\frac{\log 27}{\log 9}$, so $ab=\frac{\log 9}{\log 27}\cdot\frac{\log 27}{\log 9}=1$.